Metric map

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms in the category of metric spaces, Met.^[1] Such functions are always continuous functions. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.

Specifically, suppose that $X$ and $Y$ are metric spaces and $f$ is a function from $X$ to $Y$ . Thus we have a metric map when, for any points $x$ and $y$ in $X$ , $d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!$ Here $d_{X}$ and $d_{Y}$ denote the metrics on $X$ and $Y$ respectively.

Examples

Consider the metric space $[0,1/2]$ with the Euclidean metric. Then the function $f(x)=x^{2}$ is a metric map, since for $x\neq y$ , $|f(x)-f(y)|=|x+y||x-y|<|x-y|$ .

Category of metric maps

The function composition of two metric maps is another metric map, and the identity map $\mathrm {id} _{M}\colon M\rightarrow M$ on a metric space $M$ is a metric map, which is also the identity element for function composition. Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.

Strictly metric maps

One can say that $f$ is strictly metric if the inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degenerate case of the empty space or a single-point space.

Multivalued version

A mapping $T\colon X\to {\mathcal {N}}(X)$ from a metric space $X$ to the family of nonempty subsets of $X$ is said to be Lipschitz if there exists $L\geq 0$ such that $H(Tx,Ty)\leq Ld(x,y),$ for all $x,y\in X$ , where $H$ is the Hausdorff distance. When $L=1$ , $T$ is called nonexpansive, and when $L<1$ , $T$ is called a contraction.